Internal gear teeth



Jan. 19, 1954 I M. F. |-m ET AL 2,666,336

INTERNAL GEAR TEETH Filed June 10, 1950 4 Sheets-Sheet l INVENTORS-MYRON FRANCIS HILL.

FRANCIS A. HILL 2ND m #mw HuLL ATTY FIG. III

1954 M. F. HlLL ET AL 6,

INTERNAL GEAR TEETH Filed June 10, 1950 4 Sheets-Sheet 2 INVENTORSMYRON- FRANCIS HILL FRANCIS-A. mu -13 n!) ATT'Y M. F. HILL ET AL INTE.

Jan. 19, 1954 RNAL GEAR TEETH 4 Sheets-Sheet 3 Filed June 10, 1950INVENTORS MYRON FRANCIS HILL FRANCIS A HILL 2ND nexnI PROR ART AT T'Y1954 M. F. HILL ET AL 2,666,336

INTERNA EEEEEEEE TH INVE NTORS MYR FRANCISHILL F'R CIS A.HILL.2ND

Patented Jan. 19, 1954 INTERNAL GEAR TEETH Myron Francis Hill andFrancis A. Hill 2nd, Westport, Conn.

Application June 10, 1950, Serial No. 167,471

Claims. 1

This application is a continuation in part of our allowed application'Serial Number 561,498,

filed November 4, 1944; and of pending applications Numbers 659,098,filed April 2, 1946; and 155,176, filed April 11, 1950; and isdistinguished from the inventions in those cases by limitation tointernal change speed gears. Other internal gears have one or two teethengaging at full mesh, but lacking the circroidal addition explainedbelow, are not reinforced by other tooth contacts to enable them towithstand sudden shocks tending to break or strip the teeth. It alsoincludes features not disclosed in those applications.

Our gears have been impossible heretofore since their tooth contours arebased on. new principles of geometry which have escaped the attention ofdesigners of rotor and gear teeth.

A series of patents to M. F. Hill, particularly Reissue Patent Number21,316, described rotors, referred to hereinafter as Gerotors (theirtrade name), one Within andeccentric to the other and having a toothratio of numbers differing by one. The teeth of one maintain continuoustight engagements with the teeth of the other, during rotation. AsGerotors rotate, each tooth of one rotor'enters the tooth spaces of theother. Teeth of one rotor slide or roll over all the teeth of the otherrotor. The rotors are made with such precis ion' that contacts betweenthem persistthruoutrotationu I Gerotors were of course in sets, withratios of teeth of 4 to 5, 6 to 7, 8 to 9,. and 10 to 11.

'Ourpresent invention, under the trade-mark name Rotoids, includes setshaving tooth ratios of 5 to 7, 7 to 9, 9 to 11, etc. Such gears depend,for continuous tooth engagements, upon a difference of one andmaintained in engagement by a driving relation between them. But, havinga difference of two in numbers of teeth their actual basic ratios are 2/2 to 3 3 to 4 /2; 4% to 5 /2; and 5 /2 to 6 /2, which have thenecessary difference of one.

The teeth of a gear in the internal gear art are disposed around a pitchcircle, the radius of which is arrived at in substance by multiplyingthe eccentricity distance between their cenmultiplied by 3 and by 4;that is, by one half the numbers of teeth; or to put it another way, bythe numbers of teeth divided by the difference in their numbers ofteeth. The radii of the two ratio circles of 9 to 12 teeth, as anotherillustratron, were found by multiplying their eccentricity by one thirdof 9 and one third of 12. These factors were disclosed in the generaldescription and,

in the drawings.

In our present invention the same rule applies to our ratio circles sothat the radii of ratio circles for five to seven teeth are found bymultiplying the eccentricity by one half of five and one half of seven,or 2 and 3 which is a basic fractional ratio having the difference ofone, an essential factor of continuous travelling tooth contacts betweenthe rotors. For a ratio differing by three, 5 to 8 for example, theradii are the products of the eccentricity times one third of 5 and onethird of 8, or 1% and 2%, which also is the basic fractional ratiodiffering by one, essential for continuous travelling tooth contacts,all described and'illustrated later in the drawings and specifications.

These ratios are not variations in degree from each other or from evennumber ratios such as 6 to 8, 8 to 10, 10 to 12 and so on. Ratios ofeven numbers of teeth cannot be generated at a single setting butrequire that half the teeth be generated first, and then the blanksindexed to generate the other half of the teeth between. This nearlydoubles the labor cost. Manufacture requires tolerances, often of two orthree thousandths of an inch. Such an error in indexing would result innoisy gears.- Even a small fraction of this tolerance would result inuseless gears for high torque loads. It was the addition or subtractionof one tooth from the even ratios that made possible the generation ofall theteeth at one setting. Such uneven numbered ratios are thereforedifferent from the even ratios by more than mere degree. In presentcommercial sizes 5X7 teeth (2 /2 to 3 /2 ratio) haven't enough space inthe pinion for ashaft capable of operating under high tooth loads.

The 7 X 9 teeth (3 /2 to 4 ratio), same eccentricity, allow for anadequate shaft but not for an adequate heavy duty roller bearing insideof the pinion,.needed for some uses. But 9 X 11 teeth (4 /2 to 5% ratio)same eccentricity, do

teeth), same eccentricity, provides for telescop- I ingof bulkheads asin M.,F.' Hill et a1. Patent Number 2,484,789. This basic fractionalratio 3 differing by one mares smaller gears possible. For instance, inan 8 by 9 toothed pair of Gerotors with an eccentric distance of betweenthe center of one gear and that of the other the radius of the 8 toothpinion ratio circle is s 8 inches. The radius of the 9 tooth outer gearis 9 *716= /16 inches. In a pair of Rotoids having the same 71eccentricity and the same number of. teeth the outer gear, the outergear ratio circle radius is 4 ,i inch while the 7 tooth pinion ratiocircle radius is 3 A inch. Thus we see that the Rotoid gears areapproximately half the size 'of; the: Gerotor rotors. Hence it appearsthat no difference merely of degree, existsbetween gears having suchnumbers of teeth.

As to gears of the same eccentricity; having-,the difference of one, a 4X 5, or 5 X 6 ratio is not so limited as to shaft size because ofshallower tooth spaces. They have poorer pressure angles andla'ck-the'displacement'ofxthe-fi% to 3 /2:r'atio;- Similar difierencesoccur inratios of larger nurn bers'i High pressure is limited by th'ethrusting of the teethagainst' ea'ch otheron the low pressure side where1 frictional rub between the teeth, greatest toward open mesh, createsheat, gumming of'lue bricating oil and binding:

Rotoids have no tooth contacts-across: open mesh or on the'low' pressuresidegso thatpr'essure; cannot cause their teeth to rub; create heatandbind. The-only mechanical pressure is that n ofone driving the otheracross -full mesh with a rolling actions The-open crescent range at-openmesh, assisted by: back lash, is 1 the factor that prevents the:teethLfrom rubbing on each others Itmakes possible steady speeds attoothpressures rangingintothousand's of pounds.-

In' our Patent2,386,896 the -off sides of adjoin":- ingteetl'i ofthesouter gear are=:limited:to -a c'o1nimon center. Rotoids have no suchlimitation.

AsRotoids rotate at ratio speeds,-.onewith re lation. to: the other,that is; V at speeds: inversely proportional to their numbersofiiteeth,--our' new contours make similar I continuous wipingf and g5rollingcontacts witheachotherz.

Departure from the: Gerotor methodiof de signing .gear. contoursimakesfpossible thehunting: relation. betweenrallithefite'eth of the:gears and:. improved pressureiangles. in the drivingtrangesbee tween theteeth atrfull mesh.

The crescent range. also; prevents the; pinionz. fromlriding ontheteethofithezouter: gearufrom oneend of the crescentdao therotherz'This ares;- centrange. isatopen mesh betWB6n.l3W0.- 0V8I' lappingcircles, one along thetips of.thesteeth:o'f.. one zgear andsthe. otheralong the tips.of.the'teeth ofthe other. gear.

Backlash was .old, .butzcombinediwith.theicone tinuous contacts, itprevents .theteeth' of'thezpine 6Q ionfrom riding on the. teethof: theouter: gear fromone-end ofithe .crescentzrange.tofthedriving range. atfullmesh, -while leaving the continuous contacts between the. other endofithe crescent I range and full meshnna'ffected; A'zdifierencaot twoor: more teeth;between gears}. old: in themselves, .createdunexpected.results-with. continueous contact gears, causing-the rule ofcontinuous: contact to. be modified, firststo.includemultiples. of: the.ratio having a diiference of Lone; and 1311811170 changed to include thehunting relation;

It also: permitted gear teethvt'o. bewbuilttloose, with thexcont'acts.brought into' actionsbydrivingz Thusloosely assembledgears maylapJthemSel-Ves to. a perfectlfit 1 in servicezwithouti binding;.-

These step by step developments, each valuable in itself, produce newcharacteristics in continuous contact internal gears.

The first difference of two teeth appeared when, inspecting Fig. IX ofthe reissue patent to M. F. Hill, No. 21,316, the tooth spaces of thepinion appeared wide enough to hold another tooth, and the outer teethwide enough for another toothspacel. This;doubleol'thesnumber of teethand they acquired a difference of two which provided the much neededcrescent space to eliminate the riding of one gear on the teeth of theother'at open mesh, without loss of the continuous contacts. See, also,Patent 2,386,896, Fig.

Useful in reduction gearing this idea was patented in p,atents,-- Nos.2,091,317 and 2,209,202 to M. F. Hill. The disclosure in these patentswas limitedto multiples of teeth having a difference of one. The contourand pressure angles of the reissue patent, 21,316. characterized the.teeth.

Furthermore, in'rnultiplying the teeth, a, part of the tooth height wassacrified; and whatever.

wasdone'to. them thereafter, such'asreducing thenumbers ofiteeth againwhileincreasingttlie eccentricity, the lost height still: prevailed;The? height of atooth affects the size and durability; of. the teeth.Incidentally. the centersv of the curves of two. sides of each'toothwere doubled in'i-numben an'd were evenly. indexed. with eachother. Inour Patent No'. 2,386,896'the offlsidesnf the adjoining teethof theouterz'gear had to have:

a common center, which aflectstherelation between the near. sides I ofalternate pinion teeth.

Ourpresent gearshave no such limitation.- The: multipled gears hadthexcontours near thecon Vex tops or crownsthat the original rotors-'had. When teeth are multipled, even'tho theyyhad a hunting relationbetween them, the'huntingrelation betweenall the teeth is lost.

Gears having, a difference of two: teeth, result ing. from: multiplyingsmaller ratios oftintegers differingby. one, do not have thehunting-.rela tion soessential to easy manufacture and good. service. Toindex two sets of teeth withrelation. to each other in a machinecutting. them,

with theexactness needed for efiicienttight-.relations, if: notimpossible,- is at least a great ex pense, a'nclnin productionin.quantitydifiicultto maintain. In generation, one set of teeth isfirstgenerated on a blank, theblankmust then be indexed exactly half. way,and the second-set generated; Error results in hammering'of teeth,noise, lost: efliciency, and:wear.

cient service.

With full hunting relation there is nossecond indexing, and exact:indexing; of all the teeth atone. setting is accomplished. The huntingrelation-.enables-each: tooth toengage. in proper turn all the teeth ofthe other rotor: Thiseevehs up wear due to differences in hardness; ordura moving a toothhas thesameefiect: til-X gears:

mightbefalteredito '7 X- 9 or9 X:.11, vfor example.-

Then came:v another: conceptiom, the Hill Theorem, a more scientific:method =1 of: designing" gear!contours. for. continuous contactparticu* larly useful .for gears havingdifferences in.num.-*

bersofateeth greater than.one. A difierenceoi The usual marl- Fuzfacturing tolerance wouldruin gearsfor efi'i-w twoincreased the heightof a gear tooth, and added displacement It made possible a betterlocation or inclination of the sides of gear teeth to get betterpressure angles. ;It, made possible the location of drivingsurfaces'across the ratio circles instead of mostly outside of them toreduce angular slip. It made possible the more intelligentlaying out ofgear teeth, selecting the best radii of curvature, and determining the,'Circroidal Addition more accurately than in the generator machine,which had been the best means for designing rotor contours before.Mathematical calculations are intricate and take much time. But in agraph, comparison of different curves is easy. It is the final step inadapting this new type of gears to commercial use. .In new geometricalrelations rules or laws are sought to guide those to become skilled inthe art. The first rule that appeared important, was that in gears, onewithin the other, there must be a difference of one tooth only, since atooth of one had to travel continuously over all'the teeth of the otherand the speed ratio had to be the ratio of the teeth. This was acceptedfor years by those developing and commercializing this art. The initialpatents, to M. F. Hill contained claims limited to the difference of onetooth.

When it was discovered that the teeth of a ratio differing by one couldbe doubled, to make adifference of two, or trebled to make a differenceof three, the rule had to be modified, and extended to include themultiples.

.Thusit was discovered that it was not the ratios of teeth that werecontrolling but the basic ratios themselves which had to have adifference of one, even if ratios of fractional numbers were necessary.

Then it appeared that doubling a ratio differing by one lost the huntingrelation between all the teeth. The even numbers of such a ratio had tobe made uneven, possible only for teeth based on a fractional ratio,having a difierence of one, the actual number of teeth being found bymultiplying the ratio numbers by the denominators of their fractions.

The lowest such ratio having a gooddriving relation between the gearsmay be 2 to 3%,,

the gears having 5 X 7 teeth. They are not multiples of any ratio havinglesser numbers of teeth, and have larger teeth than other ratios ofhigher numbers.

A 1 to 2 ratio, with 3 X 5 teeth is possible for low pressures, tho thedrive action is poor.

A to 1 ratio, having 1 X 3 teeth requires additional driving members orgears to keep them registered. Multiples of very low ratios,-

1 to 2 or 2 to 3, doubled or trebled, on a shaft,-

may fit some needs; all having the continuous tooth engagements.

With the pinion drive, so called as in Gerotors, a pinion is freelymounted on a drive shaft and key, to find its own best running positionwhile driving the outer gear. With 5 X 7 gears the tooth spaces are sodeep radially that they almost reach. a shaft, of a size needed forheavy torque load. Such teeth are integral with the shaft. But a 3 to 4ratio leaves room for an adequate separately made shaft for high toothload.

A difference in numbers of teeth of a tooth ratio is not a matter ofdegree for another reason. In rotorsof the same size, a difference oftwoteeth increases tooth. height while a difierence of three'reduces it,because of the longer crescent space and fewer teeth at work together.

Enough backlash to prevent the teeth on both sides of one gear fromriding on the teeth of the other opens crevices between the teeth,unless prevented by the drive. The end of the path of contact occurswhen a crevice between contacted teeth starts to open. The location of atight engagement over a range from one tooth to the next is incident tothe ending of the path of continuous engagement.

Paths of continuous engagement vary in different types of continuousengagement gears. There are hosts of types based upon the geometricalprinciple of the Circroidal Addition. Of the three principal types ofgear contours, the Gerotor of the prior art, type, having a differenceof one tooth, has all the teeth in continuous contact with each other.The outer rotor has a circular curve on each rotor tooth extending overthe top and sides with a common center, the tooth centers of all theteeth being evenly indexed. The second type is that shown in thegearPatents Nos. 2,091,317, etc.

vex crowns. The are centers are evenly indexed.

The third Rotoid type has teeth differing bytwo or more with 'a fullhunting relation, with a center of the arc on one side of each outerrotor tooth unevenly spaced between the centers of the arcs on the otherside of the teeth, afiording larger radii, better pressure angles andgreater size.

The paths of tooth engagements differ in these types. In the first orGerotor form, the path is continuous around all the teeth, with a loopat full mesh. Without back lash in the second and third type the pathstarts from the end of a crescent range and stops just short of the.full mesh point.

Another path is the reverse on the other side of the gears. The thirdform, in this case, is complicated further by the odd ratio based onfractional numbers having a diiference of one, which has the huntingrelation. The location of path of contact on the side of teeth notdriving,

in the second and third types, is removed.

Pressure angles between the teeth in the driving range have an importanteffect upon the internal resistance of a gear mechanism and the power todrive it. The 30 to 50 angles in the 6 X 7 Gerotors create resistance torotation by thrusting the outer rotor away from its axis and thrustingthe teeth at open mesh against each other. In the gear patents, No.2,091,317, etc.,

the tooth contours and pressure angles at full Pressure One of thefactors of the low internal resistance is the character of the toothengagement in the driving range at full mesh. The resistance in Rotoidsis reduced to a pure roll of a convex The outer teeth have the. sameside arcs of circles, but no complete con- The radii of curvacurve-2on6another curve having: slightly'larger ratfii curvature; m 7 x x-'91rotors: having": thesamtype of f c'ontoursi the' action is even nearerap'u ol ll mucus: contact"- rotors o'r gears are: now

has i eircular cuttingie'dges to cut the circular arcs on the outerteeth, The bottoms of the tooth spaces; do not have to have agenerated-=- form as long as they clear the tops =of the pi1iion teethduriugrotation;

The Ppin'ion' iis'; generated by means of a tool, be

it a" millingvlcutter or grinding wheel; which: the? shape 0f 22 toothof the outer" gear; in this larger; this tdoth cas'eguof circular arc;is,=;the flarger. thei to'ol. and; the faster its operamm. The relative"size shown in i Figs: I-IILwas selectedas a means between extreme sizes"eaeh' of which'in that-form' has a disadvantage; in reducing tooth area;

mechanisms having. continuous: tooth ening r'otationpand run fasten andfaster: The power" consumed, high at' first, steadily drops untilwhen:completelytlapped {to each other, they may be run at high speedswithout generating heat;.v

In Rotoids having the crescent: and i back lash,

the pinion c'a'n'drop thru) the outer gear without-touching it. Withoutthe drive that brings the teethtogether tight engagement is laching;

W-iththesameinumbersof teeth and the same eccentricity spaces of teethmay vary the path ofcontact and the gear tooth-size; In Gerotors anycontour of either rotor, convex or"concave;

Fig. (I shows '5X--7- Roto'ids of a select-ed ecoenw tricity'(1 '-of.arr-inch) integral with the shaft f'on high tooth pressures.

Figs. II and III show them mounted on a smaller shaft -forlow toothpressures:

Fig. IV isa" diagram of the Hill Theorem.for epigeneration of-5 X 7Rotoid lZOOtllfiOIltOl-HSr- Fig. V illustrates-howa 5 X 7- Roto'id curveis arrived, at, usingthe' diagram in Fig; IV;

Fig. VIshows aset of -7-X .9 Rotoids-ofithesame eccentricity; and how itallows for a-large separate'driving shaft for heavy loads, with completecircular tooth crowns on the outer gear;

Fig. VII-shows-a '7. X9 set of the'same'eccentricity. with teeth ontheouter gear of. larger radius.- 4

Figs. VIII-and IX illustrate the Hill-Theorem for this-7X 9 ratio.

Figs.- X and-XI- show the method. of: designingcontours of 9 X 11 toothRotoids:of:- ;ecc'entricity, theincrease in. size being. needed forclearer. illustration.

Fig. XI I. shows, on a larger scale; the Hill Theorem for designing 9 X11 Rotoids with a hypo system of generation.

Fig. XIIIshows l-X 5' Gerotors of 'the prioi'art;

Fig, XIV includes a'section onlineXIV-JILV "manufactured: by breachingthe= outer rotor: andgenerating? the pinion: The broach' 8. Fig.Wihcludes a section of -'F-ig. ch -line Difi em;gear teetn qontourshavecorrespond ingly differen t pathsof tooth erigagementsi The contourof a pinion is' determined by generating it -usin'g as a generatihg toolacutter or grinding wheel having theiorm and size oi a tooth 0f the thataxis -is rotating 'aroun'd' the axis of theoilter" of-whi'eh the tooth-lis a pa'rt. 'Iherrthe cuts or" grinds" the tooth contours of the pinionteeth; starting into a first tooth= space (less-rip or ciitti-rig one'side' a'sut enters; andftii'ej othe-"r' sideas it 1eaves, then passihgto-the thiid tooth space (skipping the second), forming that" tooth=sp'ace'; then,-- shipping the femur and passing to the fifth thenskipping the first, passing" to theseeondand so on to thefo'urth whenthe gear is completed. This unusual genera-ting process resultsfroihunusual tooth: ratios and their ratio-circles;- The 5X-7 toothgearshaye a=ratio which differsbyone; anecessity'forcon tihuous'contact' gear teeth contours;

The r'atio -is-= not between integers-but between. fractional numbers,this case; 2 /2" and" 3 /2; But sineegears-cant run with a half tooth,these numbers are multiplied by the denorninatorf of" the fraction,th'at'is byj2 providing thenumbers of teeth oesix v. fif thefr'actionalratiofwas to" 3 k; tli'e tbdth hlimbe'r s' would'be xi 13 theratio'numbers multiplied i bythe denominator 4; These fractional ratios havewhat" is called a "-hunting relation'by which each tooth'o'f one gear'in' som'e order; makese'rigage'ment, a travel engagement} witheverytooth of the" other gear. That makesit' possible" for" a toolrepresenting one tooth of .one gear to cut everyto'oth offth'e othergear? When the toothforms of" one gear are selectedorforme'darbitrarily, the bottoihs of the" tooth space's may be" of minorimportance',; pzirticularly on the *outer" gear, and as long astheloott'oins keep out'of contactwith'the' tops of theteeth'" of the'other'gear, particularly" thepin'ion teeth; the" depth 'orjformmf the space isunimportant, so far as continuousengageinent is: concerned: Reference ismade to "the reissue patent"to-*Myr:on Hill; No: 21,316 for" furtheron-of the generating process and tothe" patent to Hugo" Bilgr'arn and M.Hill, No;' 1,798LO59 describing a' suitable generator. For thesecontours the mill or grinding wheeltravels' ac1-*b'ss* the"gearblank,ina" direction parallel to the gear axes; accomplished, by. mountingthe" Bilgr rn' l iill machineupon' a"shap'er, which may carry agrindingheadf Gth'er ratios m'ay-be provided, such as"'3 /2"to 1. /2 fle' tdh/2";etii: or 3% m4 /4; or any other fractional ratio differing by one?Only these fractional ratios have the desiredhuntingrelatiohi- Thconvexflporti'ons oi" the" teeth. of the outer" gearareeircular"arcs?They need'ri'ot be circular arcs' since other contours may 'be used: Anyper tier-1" off an ellipse; c'y'oloidi; parabola hyperbola'; irregularoval or mongrel shape; may" be used iristead o'f circulaharcs;Whateverfor'mis used; they impress; durihg'generatioh;theirc'haracteristies' up'onthe contours of" the "other gear; and

exact mating curve or it may be deeper.

desired upon the tooth spaces on the pinion.

Generation may be assisted by our Hill Theorem, so called, now taught insome leading scientific schools. It is illustrated in differentapplications, in Figs. IV, V, VIII, IX, X, XI, XII and XVI.

In Fig. I, the pinion I has five teeth around the axis Y, the sidecontours of the teeth meeting at the tips at an angle. The outer gear 2has seven teeth, each including circular arcs, 5 and 6 nieeting a pointI. The distances between any one axis 3 or 4 may or may not beequidistant from its two neighboring axes. It depends on their radii oftooth contours. Increasing the radii brings the curve axes of onetoothnearenand I decreasing them does the reverse.

While the exact tooth contours are impossible to portray in a drawing,the lines of which alone may be hundredths of an inch wide.

I The characteristics or" the contours and their method of mating areshown. In Fig. I an outer tooth 8 is meshed in the middle of a piniontooth space, while in Fig. II a pinion tooth 9 is meshed in the middleof an outer gear tooth space It, in which its contour is mated to ashape for the concave tooth space. This shape may be an Its outerportion is preferably deeper as liquid is more easily expelled. Theouter gear tooth spaces may be deepened arbitrarily, as at M, Fig. III,with-reference to the top of the pinion tooth so long as the latter hasits necessary freedom to enter and leave without interference, and itsneeded continuous engagements are not lost. I

, The diflference of two teeth creates the crescent space II at openmesh between the broken lines the tips tooth engagements is indicatedinthe dotted line I5 extending from the left end Id of the crescent IIto a loop nearing a point at It, near a center line thru the axes Y andO and then on to the I point II which is the right end of the crescent II.

Some gears are more serviceable without back lash. Other particularlyfor high-tooth loads would be forced into heavy tooth engagements,

with teeth at I8 and I9, to the detriment of their tooth surfaces. Suchgears are provided with back lash, as in Fig. II, where no contactoccurs at 29, 2i and 22, on the right side of the' two cenj ters Y and0. To create back lash, one method I is to generate one of the gears,the pinion for example, then index it slightly (in either angulardirection) and regenerate. This shifts one side of a tooth towards itsother side, narrowing the tooth." The same shifting of one side towardsthe other may be applied to the teeth of the outer One or both may beemployed, but one is If the pinion in Fig. I drives clockwise gear.enough.

, or anti-clockwise, engagements between all the outer gearanti-clockwisethe teeth engag at 24,

, 25, and ZG-as they open. H V Y I is endless.

In Fi I the to I6.

In Fig. II the path of contact Rotoid path I5 extends from I4 29 withanticlockwise pinion drive ranges from 24 and 25 to With clockwise driveit would be a similar path in reverse on th right hand side of thecenters Y and 0.

Fig. III has the same paths (not shown) as Fig. II, with the gears in anintermediate position.

Our invention is applicable as to some of its features to many constantcontact ratios. The 5 X 7 ratio is unique in that it has the fewestteeth of the fractional ratios, its displacement requiring for mediumfluid pressures, a driving shaft so large that the teeth have to bintegral with it. Its pressure capacity and low pressure angles make itsuperior to both the 4 X5 and X '7 ratios.

Our earliest efforts in rotors, indicated inPatents 1,682,563-4-5, werebased upon the hypo system of generation, where a pinion tooth form,preferably circular, was used as the master form by means of which therotor contours were designed by generation. The shortcomings of suchrotors were expensive and necessitated the reversal of the method, sothat a master form was the tooth form of the outer rotor, stillpreferably circular. Th main advantage of the circular form is thecomparative ease of making tools, since other forms require special andintricate mechanical equipment.

Hill theorem tours for comparison, to estimate their relative values. Itcomprises an outer ratio circle B or arcs BA of it, a pinion ratiocircle A, a radicroid R which extends from I) thru Iii] to IIIII] and acircle of eccentricity E.

A circular form may be pivoted at the tip of the radicroid to generate apinion contour, but other forms may be used. If a non-circular form isused it is fixed to the radicroid, not pivoted. The pinion circle A isdivided into equal numbers of arcs. Ten are convenient for a five toothpinion, some of which are marked 00, II, 22, 33, 44, 55, 65, and II,others not to be used. An extra division between 22 and 33 assistsaccuracy for reasons to follow. It is marked 25. The

eccentric circle E is also divided into ten parts,

I), I, 2, 3, I, 5, 6, and I, etc. The others are not used. The startingposition of th radicroid is from the point II in E thru the point tillin A, and on out to the point IIGII selected for experiment. The ratiocircle of the outer gear is not drawn in full to save unnecessaryconfusion, and is indicated by the arc B in the starting position. Asthe ratio circle rolls it assumes successiv positions indicated inbroken lines BA. As the ratio circle rolls, and assumes these variouspositions,

' its point III! travels to various points Ii), 2e, 25, so,

4t, 59, 6D, and I8, which in reality are pointsin a cycloid, since apoint inone circle rolling on another travels along a cycloid. Meanwhilethe excesses 111 reenter iof thiswatio circle, point Win the circle ofeccentricity "travels "around this circle E :ithru lathe various zpcintsf L5 2, :etc. ."As :the radir'croid R coincides with and includes Lthera'dius of B, and moves with it,zits 'tip,:.tout.tbeyondfthe ratiorcircle, traces a curve, which 'runs' thru the ipoints 098,199, 230,255,390, 400;59135600, \TUU etc., or as many others in between asmaytrbe needed for accuracy. 'This is'thecircroid wanted for trying outgear curves. Another circroid might'be ORA. 'A circroidisa speciesoftrochoid limitedito generating "gear'curve uses. "Pointilllm isselected arbitrarily as a starting point. Varyw ing'itgvariesrresultantcontours of teeth. As B iro'llsanti-clockwise on A the DOiIIt UDIXieSCfibBS a cycloid C. As 09 travels to the left along CR; the'radic-roid is carried by the circle B) with it. The pivot o'f theradi'croi'clfllsalso travels around .E anti-clockwise. Thus theradicroid R in:each successive position has two points 'to locate it'and ithll$-itS l1teI' E1'l d Q59 traces a definite 'path CR. YWhen B,rolling 011 A, istangent'at 1,?0 reaches in -E ;'&0Q:rea ':hes I l in Ayan'd ace reaches Hill in \'CR,*etc.

V' transfers from Fig. IV; the pinion ratio (circle A; circroidCR;eccentric circleE and radi- =':cr0id R, for plotting the criticalinner limit 'of the pinion contour. The figure shows this cir- "croidiCR 'and the pinion ratio-circle 'A with its above described dividingpoints. -W .seeka-conitour parallel to or 'equi-distantfrom the choroid."The :scircular are is 'to be used to "describe this -=parallel curve;its envelope; from successivepoints :*along' the circroid. We "do notyet-know the best length ofthe radius of M. "If it is'too longwacritical'port-ion of the envelope MBwvill be broken into .parts crossing-at different angles. There :must 'bea criticalpoint inside ofvvhichtheperf'fect geartooth contour rnust l-ie. 'If'it is tobe parallel :to Pthe circroicl it must have all its normals-i. e. normals to its tangentsalsomorm'al to the envelope, and for.an.equi-distant envelope all suchnormalsmust be of equal length. If ..-the-.curve .MAisanon-.circularcurve, theen- .-velop.e. corresponds: to its irregularity.But-.inthat -.direction-.lie complex tooth. forms .with .circroidaladditions torcorrespond.

-The radicroid includes the .portion .0 to All] rwhichiis the radius ofthe. ratiorcirclerB cf .the

outer gear 2 in Figs. I-zIII. llhis-ratimcircleis .shown :in .Fig. .IV,tangent -.to .A. .at .its -;point 00.

.Iheanglebetween .CL. and. 8-69 is.-36 which -.is -.one-half of atooth-,division.of. the gears.- in..F'igs. Ia-III.

:These .locations .are .selected .to .conform :to I

f'mathatical formulaeior points ina gear. curve, :in terms oftheirordinals andiabscissae.

If rthescircle-B, tangentrat 09, startslto 2110115150 tthe .left, itspoint .of tangency .travels .along .A .thru the successive points .-AlI, A22, wA33 -tc.

tthelibroken.linesBA. Thepoint -00 ;in the radircroidi. .e...the-.-endof..the;radius .2fl-=8 0, travels over :acycloidCLthru .0 L0,. C2 0; C30 etc.,.to ,points". in a curve .CR,...designated at .CRMlll, .CRIUO,.CRZOU, =.etc.

.The corresponding -radicroid .positions are .sketchilyjindicated by.brokenllines .'from E .thru .CIU, ..E2, -to C20, .133 .to C3.0,.etc tothe same gp0ints:.CROU9,'.CR [[90, LCRZOB, .etc. These broken..lines.therefore-extend outward a distance which :is :kno-wmasthe.circroi'dal .addition'." This disttancelis arbitrarily vchosen. "It isa critical fac-- .tor.

jcycloids .-.have .instant .radii "down i'to rzero in 4 length, hence noother curve inside can be paralcircle. B at :these locations isillustrated .in

H2 :zllrorequidistanttfromzit. Butithisiszndtztruemf rthercircroid.Whereitheaeycloiolrhas:a;minimum aradiusaof zzero, za circroid :has a:minimum .of .90".

aAs :the :radicroid atravels, this 9.0" ;angle .is :in- :5 .creasedup'to a maximum, and permits-withinzit, zcurvesaequiedistant :or.paralleditoit. .These are ethe :curvesgofxpossible :gear :teeth, :and:are :-to .:be :laid :out.

sIn FigaV .therez-is azcriticalrpointrb etween which and :Y .a'toothcontour :forrcontinuous contact :iszimpossible. ,:I-t.:'ha's1to::be;located.

Hnstantzradii of the rcircroid rare rexplored to f'find :an intersection.:nearest to :the circroid. .These'radiizobviously originate on thecirclerA upon which the zcircle 1B .is tangent, candia ,l'me .ifrom thistangent point .to the tip. of theraolicroid, rinzthe circroid'GR'isa-aninstantradius. Instant :rradiiirom {points-iGRZDWand-CRSM .to pointsJ 2randsflzof thecircle A intersect;at.N. If. another ;.1instant radinshalf way between .:-A22.CR200 eand.ZA3.'3zCR3l00.- is plotted; it.alsointersects. N-ior rvery:close:to:it. If aqgearcurve is locatedzon-Nit has a sharp corner, whichtisto be avoided i'fOI .useful pgears. .A'r-adius shorter than .-A2.2-.CR200; is'therefore selected .as: of. a.circular generating form Mato; outline anrenvelopewhich is'ttoybe a gearicurve; Otherlformsreare.contemplated. of which"; more anon.

The -form.MA1may.- be. a :tool form for .cutting .-.a gear blank(with-its'stroke; par-allelrto the-axis :Y) .orritimayberan arc struckbycompasses. from r-the ;.point: CRMO ras a1 center. .Thisztool form.- orrare travels .along .c'entered at successive .points :of the choroidC-Raoutliningthe. envelope MZ-use- :ful-for. a.tooth..contour. ,partof.the en- .velope lyingmithin 36 may. he. one side. of .atooth.zRadial-lines .from' Y vthruAClll and A H are .36" apart," also v'thru..Al- I (and AM. A- curve .from 3 on .Y-AH -.to .T..on =YA22 "is.therefore suitable asrone side .of.-a gear .tooth. .That portionof. the

.'.arc MA.that.outlines.the contour-.MAS. to Tie employed as acorrespondinghalf.of.the-.outer.gear tooth its other side -;beingthe-same .in .reverse (usually).

- J ustas .a; radius: of a-cirole .is .normal -to. it, .so ncircroidshave radii .normal .to tthem. Such a radius. is-.from 1a; pointof.thecircroid .to .a corressponding; point. on theratio circle A-towhichvthe ratio circle .13 =is .tangent at ithe .time. .The .end :of the.radicroid, while .tracing the.circroid, is :-.swinging 1 or .turning...on .-a travelling .point-ethe point..of.tangency between the circlesAandLB. A; few of these. instantradii are. indicated. in this :Fig.-.V,'..one..from.2ll0 .to. 22,. and another from 300 .to .33. .Theyconverge -=-more than any I of the .other instant radiifromnther.points. Another ..from.250.to .2 5..also inc1inestoWard-3 00 .to 33,even .more. .That pointof intersection between these various. instantradiinearestto. the circroi'd ,1 is the .zcri-ticat pointtha't weare'aftergsince any envelope .beyond it is broken. uplby. the. arcs;lying at inter- .-.fering angles. 'lThisgpoint oflintersectionisin-.dicate'd :at ,N. .JAny envelope between 'N and the .circrdid (radiallyoutside 01.N) will provide a ltooth .curve having. a continuous tooth.contact -relation with ato'oth curveMA'ofanouter gear. Thedistance orthe tip of the 'radicroid toits ratio circle istermed the fcircroidaladdition, and with "a "given ratio, this circroidal addition'ideterminesthe distance of N from it. Circular mastert'formS-"have beendescribed, but any tooth "curve de'signed for gears having radii ofcurvature greater than the zero of a cycloid, whereit:crossesthe-ratio.circle, must observe 'the'require rment'df- -the*circroidaladdition" in order 'tox'maintain continuous contactsatzsteady;speed. =(The same is true as to such tooth curves, the tops ofwhich are cut off at the ratio circle as in certain gears.)

The nearer the tip of the radicroid is to its ratio circle, the shorterthe radius M. By such reductions the instant radii of the circroid arereduced. If this reduction is carried to its limitzero--the circroid ismerged into a cycloid as C, and the radius of MA is correspondinglyreduced to 0. To put it another way, there is no envelope possiblewithin a cycloid and eq'ui-distant from it. It is the failure of geardesigners generally to observe this fact that is responsible for. noisygears and limited durability.

One would naturally suppose that in order to generate a tooth of a gear,a blank would be located on the inner ratio circle axis, and a tool togenerate with located on the outer ratio circle The generating toolcertainly could not be carried on a greater radius than that of theratio circle, they argue, because the speed ratio would be changed. Thatis where efforts to solve this rotary problem undoubtedly bogged down;It is the illogical idea that solved the enigma. For while a master formis mounted on a radicroid of greater length than the radius of a ratiocircle, nevertheless the resulting generated tooth contour may lieacross the ratio circle as gear teeth should, and thus travel at thespeed ratio. It also might lie outside of the ratio circle but thatlocation introduces angular'slip and poorer pressure angles.

A rotoid tooth may have difi'erent arcs on its two sides. If a masterform is a composite of different radii of curvature, the point N is tobe determined from the several segments. which lie nearest to thenearest circroid. Contours once determined may be modified or undercutwhere not needed for engagements.

In order to locate as accurately as possible, the intersection N in Fig.V a large chart was used, and instruments of accuracy located thevarious points involved. The relative location shown is' approximate. Anumber of normals were drawn from points on the circroid between 200 and309 to corresponding points upon the ratio circle A before the locationof N was finally accepted. Varying the ratio or circroidal additionshifts it. For the relations shown in Fig. V the intersecting normals orinstant radii of the circroid are between the 288 and 300 positions. Forother relations it is nearer the point @011 or further away from it.Some of the normals diverge and require no consideration. While thediagram method is a shorter and easier one, j differential equations maybe used to find the mathematically correct N, by increasing thecircroidal addition with the same M radius.

Practically, the tooth curve is drawn as an envelope outlined by arcshaving the radius M, centered at successive points all along thecircroid. The radius M must locate theenvelope between the point N andthe circroid. The nearer it is to the circroid, the more it partakes ofits degree of curvature. The nearer it is to the point N the sharper thecurvature around N. If carried to its limit, a corner of N is arrivedat, too sharp for use. By having the curve located a small distance fromN as indicated, the best results are attained.

Pressure angles are involved in the radius of the curve M. The less theradius, the greater ing range.

' such an angle of 36.

the variation in the pressure angles in the driv- Average pressureangles for a given ratio are derived from the largest useful radius ofcurvature. The inclination of the curve is also better with a largerradius, due to the curve centers'being farther away around the ratiocircle, as indicated in Fig. IV where 000 is to the right of thevertical axis, while the tooth curve MB is to the left. The pressureangle is equal to that between a tangent to the driving curve at anypoint, and a radius of a ratio circle to that point. During a drivingrelation over a driving range of one tooth division, the tangent pointis traveling, hence its angle varies. A fixed pressure angle barscontinuous tooth contact in the driving range.

As the circroidal addition is reduced as described, the critical normalfrom the circroid to the ratio circle is ever shifting along thecircroid towards 960. With changes in the numbers of teeth, and withvariations of the other factors mentioned, the location of the normalalso changes in one direction or another, and'in designing differentgears, with difierent relative radii of curvature, these various changesare studied to select the forms most suitable for the gears desired,compromising upon numbers of teeth, size of master curves, circroidaladditions, for strength, low pressure angles, and size. By making graphsof the effect of the changes of each of the factors, one is enabled toselect more intelligently the form best suited to the problem in hand.

In order to design 9 and 11 tooth gears the ratio circle A2 of Fig. Xhas a radius 4 /2 times the eccentricity, and the ratio circle B2 has aradius of 5 /2 times the eccentricityf Circle A2 has a number ofdivisions laid off on it of equal length and the points of the cycloidC2are located to accord with them. The rest of the procedure is similarto that for the 5 and 7 tooth gears, except that the radius M2 and thecircroidal addition have to be experimented with to get the best form oftooth curve, the best pressure angles, and sufficient tooth size for thedifferent ratio. Such experimentation consists of varying the circroidaladdition by varying the extension of the radicroid beyond its ratiocircle, then finding the point W2 for such circroid as may be describedby the radicroid, and then with a radius M2 a little short of the pointN2, outlining a gear tooth curve.

After describing a curve that appears satisfactory for the tooth ratio,its driving relation, its pressure angles, etc., the next step is toselect the portion of such a tooth curve desired for a gear tooth. InFig. XI curves were sought for a five tooth pinion. The curve T2 may beone side of a tooth. Qbviously half a tooth is limited to one fifth of360 divided by 2, which is 36. So next it is desired to find out whatpart of the curve T may be utilized for one half of the tooth. Thebroken lines GZYZ and H2Y2 are drawn from the ratio circle center Y2 atBy swinging them around the center, back and forth, they includedifferent portions of the curve T2. The part included in this figure isfrom G2 to H2. If swung to the right, the end H2 is nearer the center Y2and there is little change of diameter at G2. This might be desirable asit increases tooth area, but for many uses the shaft usable with thiscontour would be too small.

Also the teeth might lack strength'as being too thin. The teeth shown inFigs. I-III show'the ifinalucompmmise abetween these rectors. JIheytwonld ha-ve-.;to:he integralvwiththe. shait;forpurrposes .cofg-st-rength, a z uniqnevca'se. :Smaller ratios ausuaily needfoutsideigears to-ikeepithem in regisctration. iThenextlareenratiohavingthe desireable hunting relation the -7 I to =9 r'IatiO I in Figs.QVI VII. fihenext, the Qcto ,llir-ati-o, .istheone in XV. :They cover.the.:groundof-theGerotors EILOWiiI'l ruse. having iratiossof A :6; andi7, 8 I.andi9, and 10 and. l.1. The: 5= to 7 toothratiosup- :piantsHooth the popular :4 ;and --5- tooth, and the 'tfi -"anidt'iitoothrratios; now; imuse; with: .bett-er. presitsumtangies, aand zWiths greater pressure =-capacity. Else 8 .s-Xie9i rotors 're dunev frictionand increase iPrGSSYIIIB'J capacities. Ih'ere is;one-other variatien, bymeans of the hypo type oftheHill-Theoremin -.-Eig..=XVII. .Iticomprisesi'first: an. outer ratio. circle .-.=A3, rand: arcs :33 ref .:atrolling sinner rratic :"Cl'lC1. "lThe eccentric:CircleiEii,izcenteredat1Y3 I (also the :icentera'QfJAZi) isrdividediintoaeleven equal tooth:div-ision'i parts by I'POii'iIJSSAQIB, A3 I etc.,'. each eextending[.0116 zeleventhirof 1360?. :ESome i divisions a1'e:.subdivide'dastnotedriater. The rollingcircle -B3shasninesuchz'iiivisionsz'fori9'rto Iliteeth.

.AS 133 rolls Efrem its; starting point; at A3 a 1 the -:point 0.8 1 in133 describes: a rhypocycloid C3, the rrollingircircleiiheing;.alwaystangent-to -A3 passing :pointsi I $2 $3 ,tetcm and interme diate. pointsin A3.

Ascenter-i of: B3,;ate *inzthe eccentric circle E3 (130) smeanwh'ile:swingswaround 1Y3 'thru points :CII, C32,.retc. XWhenZtheroliingrcircle istangent vat I'.'in.A3r(A3l') itsicenter at'ifihasreachedthe position E3I. A line from one .toz't'ne other is an'ltinstantz radius. 1W hemtherolling circle is tangent :;ati:2.in-A35CA32) its center reaches 27632 and so Ion. 'The positions' ofOllwvhen 9 iniE: reaches'E il, H2132 etc,-.:aremarke'di I :2 etc. inthe'cyclcid C3 and iref-erred ito :as 103i, 103.2, :etc.

The circroidal: aiddition'-.in Figs.fIV, .V, VIII, :IX, .:X, "and'PXIbecomesea"Icircroidal.'-subtraction in lF-ig. XVIII; and the hypocircroid CR3: rnns"thru its points), I,.:2,f:3 .retc. iTheysarerreferredto as CR3 I, CR32,i'(3Ri33,.;etc. Hines .iare drawn from :points'A3 I,JAM A33,-i'retc. thruipointsiGR3 I (3. 3.32, 1012.33, etc.respectivelycand. extendedtoifind their ipoints of intersection.Theintersectioni Nsanearest to' thercircroidxisfla critical. point.iOther'iines Eintermediate may't'be 'dra-wntocheck .upvonsN. to rbesure-there -is no nearerintersection.

'.With .-a radius 3M3 shorter :then the distance from 'N3 tothe circroidCR3, arcs arestrucki'from points CR3 I, CR32, C1333 and otherpointsialong the curvejpassing thru CR3 I CRIB? ,=:etc. which isthe'contour MB3 of agearsought.

lBacklash is indicated 1 by the broken Cline 53 I Fig. vlI which is the"tooth contour 'beiore back dash is created. :The' contour 32 -shoWs thecentour with the back' lash. The difierence in-rpractice is sel'dom asgreat as the drawing indicates. iUsually it is a' few thousandths of.an. inch.

JIfithe outerzgear' tooth contoursdn this figure rare zgenerated .abyrorfrom the contour of 1 the "rpinionfteeth' 33, theiouterteeth end inpointsas ziintbrokenzlines 'at i34. 'lVVithsuch-teeth, the pinion'iitooth spaces zhavez'tobe deepenedias "at by .rgenerating themiwithxaitool 1 having the contour :34?xifithaouterteeth.

'When ithe=outer :teeth eare circular there vis "sthereforeeilossuoflength of path :of contact. L'Ihey tprovideraxcrescent area between 35and .31. while the pointed: outer :teeth and pinion tooth aspacesrprovide' for =a crescent of :lesser area between 38 and 38.Bilintoolsrcost. a: trifle more.

- 'jflhe-ssame-iis,truenofzthepgears'iniFig. Vi. .But iin -.F;,ig.VI'itheaverage distance from-the axis-oi rth c arsspacesriniEis. VL-is e-reatervdue tori-rarrrower outer ate-ethvand wider. tooth a-spaces.

Fig. IX indicates the difference in desigmofzthe two gear systems..The'vcur-ve. 3a9iiniFigmlXiis the curveac359 in'l ig. VII, whilethecurve in. "Fig.1 .IX is the-curvesifliimFigJiVI. Theyboth areenvelopes described in1".ig.iI Xqfrom:-the same *circroidtGR Ivwithix'sirculan curvesrofxdifferent:radii. I'Iheyv hoth lie on; thesafe='side:.of :the critical:- iritersectiOmNI The'circroid' CRIwasiconstrueted inFig."=l/YIII, *withrttheratiocircles ALTBI, theeccentric: ircle LEI ,ithe rollingzarcs I13 I .thercycloidiC. I ,iand radicroid-RL-asiinIFigs. IV and-V. These gearszit may be noted have 7 X9':teeth,:aratioaofii3%:toi4%.

' Those iniFigiXVt'haVe'Q X .11 teeth,:a 'ratioiof 4 /2 to:' 5 /2 ,ithecurves designedrasi in* Figs. XnndJXI. In this case" the pinion: ratio:circle FAQ i'has :.1-8a.di

visions, :nine of which 'i'are noted "for use. The eccentric circlei Ehas of conrse the same number. The cycloid 5C2 has corresponding:pcints. .fIhe

:ra'dicroid rnnsffrom E23 :thru 1Y2, AZMIitoaCRZB and swings around, itscenter I passingthrn E? I E22,:E2-:3, etc, its pQintAO: infiCAZ-traveling-along 'the =cycloid, and fits .tip "describing the -.:circreidCRLZL'CBH, etc. ":ThisIc ircroidis;rep,eated in Fig. XI, accompanied bythe riactors :in :EFig. X

needed for arriving-at a rgear tooth-1 contour.

Acirculariorm M2. centered ataCRZB: travels I alcng -the circroid= CR2;outlining a tooth-curve: in

it'is on line' 5". Butwits; drive .is nilras-indicated at MZ in Fig.1XI.'Whenit reacheszthe-point I,

on CR2, not shown because o'irresultingjconfusion :of lines :andcharacters, it I is" close Ito :5 -.where it ibegins to drive. 'It hasthe :best driving relation over the nextitooth division indicatediol 5A2from A2 I i to A23 '.Whi0h is oneqelev-enth 'o'f1'350 tothe angleline;near-20. :Itst'ill' hasa fair angle over the nextztooth divisionassisted :byanothertooth drive :along :'the :previous tooth division.The

- drive over the first tooth-:division has very slight :slipnnslide,being mostly ofa rolling character,

.henceit controlsawear. zAs the slip increaseson ilateritooth divisions,Wear-takes place until: there .is not enough. mechanical pressurebetween teeth ;to .cause' wear.

Angleslof toothpressure varywith. the design.

'The .lessnthe circroidal additiomthe.hetter the itzwould stop -;on-.ac.co.u nt. of swelling .from heat.

Or a drop of oil \sqnirted-intoit .would slow it -right..down. {the-.main rollers :42 .ar e separated =-byseparating rollers-A3 the ends"of which project beyond the main rollers and are enlarged to preventtheir slipping in between the main rollers. The axes of separatingrollers lie in the planes of their nearest main roller axes. Rings 45outside of the ends '44 prevent radial displacement outwards while theyare of a size to just clear the shaft 16, to prevent inward radialdisplacement. Allrunning parts may behardened. Thisis a journal bearingto run between side Walls. The outer gear is driven by a gear 4'! meshedwith teeth 48 on the outer gear.

Radial lines drawn from Y3 to A30 and A3!!! have an angle between themof one eleventh of 3602s, whole tooth division of the outer gear. Onehalf of this angle, as a protractor, centered at Y3 can define thelimits of a tooth curve, convex and concave for one side of a tooth andtooth space, after which a pinion tooth has for its two sides thiscontour and its reverse. Such a tooth contour on a 'toolrha'y be used togenerate the entire pinion contour to mate with the outer gear toothcontour. These methods may also be used with the epi ratios.

There is no limit theoretically to the highest numbers of teeth, hypo orepi, possible with our ratios having a difference of two or more teeth.The least number of teeth is determined by the manner of drivingrelation that keeps them running at the steady ratio speeds necessaryfor continuous tight engagements. provides an excellent driving relationbetween the teeth, better than with teeth having commercial forms nowgenerally used. A 3 to 5 tooth ratio has A 5 to '7 tooth ratio a longdriving contact across full mesh of a rolling character. However it hasto be assisted in part of the driving range by a very considerableradial rubbing action between the teeth. A second gear on the sameshaft, alternated, has the sheet of 6 to 10 teeth. Outside gearing maybe used to keep them in some degree of registration. Other low ratios,alternated and doubled, trebled or multiplied may fit various needs. Allthese modifications lie within the scope of our invention.

The importance of continuous engagement at steady ratio speed is perhapsrealized in connection with such low ratio gears, where continuouscontact means one contact, not two or three, and its continuity meansnothing if tooth curves are irregular, since the teeth .might haveunsteady speeds and still maintain continuous contacts. Only correctratio contours based on the circroidal addition make possible theirsteady ratio speeds.

Portions of other master generating contours such as cycloids, ellipses,oval curves or portions of them or a series of one or more of them, andV mated contours of them may be substituted for circular arcs with acircroidal addition, hence, made operable in the light of our invention.

What we claim is:

1. In a rotary gear mechanism, toothed members, one outside of,eccentric to, and having more teeth than the other the numbers of saidteeth being based on a fractional ratio having a difference of one;means to drive one toothed member with relation to the other; onetoothed member having inwardly projecting teeth separated by toothspaces and the other having outwardly projecting teeth, said teeth onsaid toothed members intermeshing with each other as they enter andleave each others tooth spaces, and having contours providing atraveling relation between the teeth and tooth spaces at steady ratiospeeds; providing for drive contacts at full mesh; provid- 'sufiicientto maintain said engagements.

2. The combination claimed in claim 1 having the contourson one side .ofsaid teeth shifted angularly providing. back lash between and incooperation with said crescent range eliminating riding action of theteeth of one toothed member on the teeth of the other toothed member notparticipating in said drive relation.

3. The combination claimed in claim '1 having a shaft for driving atoothed member; having numbers of teeth the products of a tooth ratio of2 to 3 /2 multiplied by two; and having the teeth of the inner toothedmember integral with said shaft.

4. The combination claimed in claim 1 including toothed members whosenumbers of teeth differ by more than two.

5. Rotary toothed gear members, one outside of, eccentric to, and havinga greater number of teeth than the other, the numbers of said teethcorresponding to a basic tooth ratio of fractional numbers differing byone; means to drive one gear member with relation to the other; onemember having inwardly projecting teeth and the other having outwardlyprojecting teeth, said teeth intermeshing with each other and havingcontours providing a traveling relation between the teeth at ratiospeeds; providing driving contacts at full mesh; providing a crescentrange between the teeth at open mesh where no driving action and openmesh for reinforcing each other; said teeth being based on ratio circlesor curves conforming to their numbers of teeth with the centers ofcurvature of said driving tooth contacts traveling far enough away fromsaid ratio curves to maintain said contacts and engagements at saidratio speeds, and said teeth forming tooth spaces between them andbetween said contacts and engagements.

6. The combination claimed in claim 5 having the numbers of said teeththe products of said basic fractional numbers multiplied by thedenominator of said fractions.

7. The combination claimed in claim 5 having convex driving curves onthe teeth of one gear member, and concave tooth spaces on the teeth ofthe other gear member rolling on the convex curves of the first memberacross the driving range at full mesh.

8. The combination claimed in claim 5 having the contours of the teethof one of said gear members comprising circular arcs.

9. The combination claimed in claim 5 having the dr ving contours of theouter gear member comprising circular arcs.

10. In a reversible rotary gear member, toothed gear members, onewithin, eccentric to, and having two teeth less than the other; means todrive one member with relation to the other; the teeth of one memberprojecting inwardly and the teeth of the other member projectingoutwardly, said teeth on said gear members intermeSl-iifig with "eachetl'i'r; and haviiigcofitoiiis providing a driving relation at steadym-tio speeds, providing driving contacts alt f-ull me's h; pwv minga.crescent range between the teeth 51; open inesh where no -=dr-iving=c'ontae'ts occur; prbviding teeth engagements between 'full and "Saidcrescent tan'ge fo'r riilfbfciiieht bf each other; said contents andengagements maintained tight by said di-i-vi 'ng means.

11 The 'comhirmtien claimed in claim 10--haIving the numbers at sa-idteeth 0f "said gear-memhers proportioned to a ratio based uponfractional numbers having a difference of One, the actual numbers ofteeth being the prelu'dts of mifltiplyihg each of the fractional numbersby itsdenominatbn :12. The combination claimed in "claim 1 having theteeth of the inner .ge'ar member provided with curved side contourscharacterized by an melee t eir Q i. ip

The ddmbinatienemimed inmaim 5. having for its -riunzibers of teeth 3%to 4% multipl-ied b more than one.

The cbmbinaltin claimedin claim 5 having for its numbers of t'e'e-th 4/2 to 53 2 multiplied by mere than one.

15. The combination in claim 10, having fdr its numbers 'of teeth abasic fractional ratio differing by one multiplied by the 'dendminator0f said fractional ratio.

FRANCIS FRANCIS --A. 2m).

References Cited in the f le of this patent UNITED STATES PATENTS umber7 Name Date 2,091,,317 Hill Aug. 31, 1 937 2,209,201 Hill July '23, 19402,484,789 H1117 et "a1 Oct. '11, 1949

